| L(s) = 1 | + (−1.11 + 1.00i)2-s + (0.245 − 0.551i)3-s + (0.0252 − 0.240i)4-s + (−0.491 + 2.31i)5-s + (0.279 + 0.858i)6-s + (2.00 + 1.72i)7-s + (−1.54 − 2.12i)8-s + (1.76 + 1.95i)9-s + (−1.76 − 3.06i)10-s + (−3.23 − 0.743i)11-s + (−0.126 − 0.0729i)12-s + (0.304 − 0.938i)13-s + (−3.96 + 0.0910i)14-s + (1.15 + 0.837i)15-s + (4.32 + 0.920i)16-s + (4.87 − 5.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.786 + 0.708i)2-s + (0.141 − 0.318i)3-s + (0.0126 − 0.120i)4-s + (−0.219 + 1.03i)5-s + (0.113 + 0.350i)6-s + (0.758 + 0.651i)7-s + (−0.547 − 0.752i)8-s + (0.587 + 0.653i)9-s + (−0.559 − 0.969i)10-s + (−0.974 − 0.224i)11-s + (−0.0364 − 0.0210i)12-s + (0.0845 − 0.260i)13-s + (−1.05 + 0.0243i)14-s + (0.297 + 0.216i)15-s + (1.08 + 0.230i)16-s + (1.18 − 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0377 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0377 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.489099 + 0.470978i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.489099 + 0.470978i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.00 - 1.72i)T \) |
| 11 | \( 1 + (3.23 + 0.743i)T \) |
| good | 2 | \( 1 + (1.11 - 1.00i)T + (0.209 - 1.98i)T^{2} \) |
| 3 | \( 1 + (-0.245 + 0.551i)T + (-2.00 - 2.22i)T^{2} \) |
| 5 | \( 1 + (0.491 - 2.31i)T + (-4.56 - 2.03i)T^{2} \) |
| 13 | \( 1 + (-0.304 + 0.938i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-4.87 + 5.41i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.511 + 4.87i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.0581 - 0.100i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.73 - 6.51i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.757 + 3.56i)T + (-28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (3.37 - 1.50i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (3.09 - 2.24i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 5.00iT - 43T^{2} \) |
| 47 | \( 1 + (-7.99 + 0.840i)T + (45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-3.78 + 0.803i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-1.41 - 0.148i)T + (57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (0.232 + 0.0493i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (2.19 + 3.80i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.39 + 7.35i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.168 - 1.60i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (9.06 - 8.16i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (1.29 + 3.97i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (11.4 + 6.59i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (16.5 + 5.39i)T + (78.4 + 57.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.07803054809152929126260339074, −13.84779908798719318718493917716, −12.62338485465998986707351462226, −11.30572232056511385520787256773, −10.20591102942030720206993598169, −8.790109989063165615426904008799, −7.62452846861652273477564277869, −7.14185913884396503087761817331, −5.32857763083976774842761840673, −2.88386773753716083466850301282,
1.41337555871231145358923871746, 4.03460677714128368783717039588, 5.52299683988343647032056595085, 7.80178446670363190430566278705, 8.665055408198094296823981209736, 9.950136525383953462962943176774, 10.56228199715979984481022718092, 11.96299767393898287118101042045, 12.80126424200802369131066382509, 14.34825892884963842938220282683
